SOLUTION: Simplify into a single log term: 2/3log x + 1/3log xy^3

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Question 603761: Simplify into a single log term:
2/3log x + 1/3log xy^3

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
expr%282%2F3%29%2Alog%28%28x%29%29 + expr%281%2F3%29log%28%28xy%5E3%29%29
On both terms term use the rule A%2Alog%28%28B%29%29 = log%28%28B%5EA%29%29
to write the coefficients of the logs as exponents

log%28%28matrix%282%2C1%2C%22%22%2Cx%5E%282%2F3%29%29%29%29 + log%28%28matrix%282%2C1%2C%22%22%2C%28xy%5E3%29%5E%281%2F3%29%29%29%29

Now we simplify matrix%282%2C1%2C%22%22%2C%28xy%5E3%29%5E%281%2F3%29%29 this way: matrix%282%2C1%2C%22%22%2C%28xy%5E3%29%5E%281%2F3%29%29%29+ = matrix%282%2C1%2C%22%22%2C%28x%5E1y%5E3%29%5E%281%2F3%29%29 = matrix%282%2C1%2C%22%22%2C++++x%5E%281%2F3%29y%5E%283%2F3%29+++++++%29 = matrix%282%2C1%2C%22%22%2C++++x%5E%281%2F3%29y%5E1+++++++%29 =matrix%282%2C1%2C%22%22%2C++++x%5E%281%2F3%29y+++++++%29

And the above becomes:

log%28%28matrix%282%2C1%2C%22%22%2Cx%5E%282%2F3%29%29%29%29 + log%28%28matrix%282%2C1%2C%22%22%2C++++x%5E%281%2F3%29y+++++++%29%29%29

Now use the rule: log%28%28A%29%29+%2B+log%28%28B%29%29 = log%28%28A%2AB%29%29
and you have this:

log%28%28matrix%282%2C1%2C%22%22%2C++++x%5E%282%2F3%29x%5E%281%2F3%29y+++++++%29%29%29

Now we multiply the powers of x by adding the exponents of x: 2%2F3%2B1%2F3=3%2F3=1

log%28%28x%5E1y%29%29

And the final answer is:

log%28%28xy%29%29

Edwin